An automobile of mass 'm' accelerates starting from origin and in… - Vidyadip

MCQ

An automobile of mass $'m'$ accelerates starting from origin and initially at rest, while the engine supplies constant power $P$. The position is given as a function of time by:

✓
$\left(\frac{8 {P}}{9 {m}}\right)^{\frac{1}{2}} {t}^{\frac{3}{2}}$
B
$\left(\frac{8 {P}}{9 {m}}\right)^{\frac{1}{2}} {t}^{\frac{2}{3}}$
C
$\left(\frac{9 {m}}{8 {P}}\right)^{\frac{1}{2}} {t}^{\frac{3}{2}}$
D
$\left(\frac{9 {P}}{8 {m}}\right)^{\frac{1}{2}} {t}^{\frac{3}{2}}$

✓

Answer

Correct option: A.

$\left(\frac{8 {P}}{9 {m}}\right)^{\frac{1}{2}} {t}^{\frac{3}{2}}$

a
If power is constant

$P=\text { const. }$

$P=F v=\frac{m v^{2} d v}{d x}$

$\int_{0}^{x} \frac{P}{m} d x=\int_{0}^{v} v^{2} d v$

$\frac{P x}{m}=\frac{v^{3}}{3}$

$\left(\frac{3 P x}{m}\right)^{1 / 3}=v=\frac{d x}{d t}$

$\left(\frac{3 P}{m}\right)^{1 / 3} \int_{0}^{t} d t=\int_{0}^{x} x^{-1 / 3} d x$

$\Rightarrow x=\left(\frac{8 P}{9 m}\right)^{1 / 2} t^{3 / 2}$

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